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Polymer-Based Ultrasonic Motors Utilizing
High-Order Vibration Modes
Jiang Wu

, Yosuke Mizuno, Senior Member, IEEE, and Kentaro Nakamura, Member, IEEE

Abstract—Our previous studies showed that travelingwave rotary ultrasonic motors with polymer-based vibrators yielded limited output power when operating in thecommonly used low-order bending vibration modes. In
this study, we employ a high-order bending mode in thepolymer-based cylindrical ultrasonic motor, because thismode yields a relatively high electromechanical coupling
factor, which may lead to high output power of the motor.
Additionally, in contrast with the low-order modes with only
vertical nodal lines, the high-order mode has both horizontal and vertical nodal lines on the circumferential outer surface of the polymer-based vibrator. With these attractive
advantages, it is worth investigating basic vibration characteristics of polymer-based vibrators operating in high-ordermodes and their applications to ultrasonic motors. The vibrating body is made of poly phenylene sulfide, a functional
polymer exhibiting low mechanical loss even under highamplitude ultrasonic vibration and is activated by a piezoelectric ceramic element bonded on the back surface. The
high-order modes are optimal for polymer-based vibratorswith practical thicknesses, because their equivalent stiffnesses and masses are relatively low when operating in thismode. Polymer-based motors with the high-order modes exhibit relatively high output torques and powers, comparedwith not only polymer-based motors with low-order bending
modes, but also metal-based motors operating in high-ordervibration modes.
Index Terms—Electromechanical coupling factor, highorder vibration mode, polymer, ultrasonic motor (USM).

I. INTRODUCTIONXHIBITING simple structures, fast response times, precise positioning capability, and high torques at low speeds[1]–[3], ultrasonic motors (USMs) have been extensively investigated in the past three decades, and practically applied
to some special fields [1], [4]–[6]. USMs dominantly consist of vibrators and rotors, which are pressed to each otherE

Manuscript received November 6, 2017; accepted January 3, 2018.
Date of publication January 16, 2018; date of current version April 16,
2018. Recommended by Technical Editor J. Shan. This work was supported by the Adaptable and Seamless Technology Transfer Program,
A-step, from the Japan Science and Technology Agency, under Grant
AS2711904S. The work of J. Wu was supported by a Grant-in-Aid for
the Japan Society for the Promotion of Science Research Fellows underGrant 17J05057. (Corresponding author: Jiang Wu.)
The authors are with the Laboratory for Future Interdisciplinary
Research of Science and Technology, Tokyo Institute of Technology, Yokohama 226-8503, Japan (e-mail: wujiang@sonic.pi.titech.ac.jp;
ymizuno@sonic.pi.titech.ac.jp; knakamur@sonic.pi.titech.ac.jp).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMECH.2018.2794379

to effectively utilize the frictional force [5]–[9]. Vibrators arecore components of USMs [1], [10]–[12]. Fig. 1(a) depicts
a cylindrical vibrator, which typically comprises a piezoelectric ceramic element and a metal vibrating body [4], [9],[10]. When four voltages with 90° phase differences are applied, four flexural standing waves, u1 = Asin (ωt)sin (kθ),
u2 = Acos (ωt)cos (kθ), u3 = −Asin (ωt)sin (kθ + π), andu4 = −Acos (ωt)cos (kθ + π), are excited on the vibrator.
Here, A, ω, and k respectively represent the amplitude of thevertical vibration velocity, angular resonance frequency, andwave number (k = 3 for our motors); t and θ are the time andcircumferential angle, respectively. The superposition of thesestanding waves forms a traveling wave
uz =

4


ui = 2A cos (ωt − kθ) .

(1)

i=1

In flexural vibration, the vibration velocity along the circumferential direction is as follows:∂uz= −2αkA sin (ωt − kθ)(2)uθ = −α ·∂θwhere α is a ratio determined by the vibration mode and structural parameters of the cylindrical vibrator [1]. Thus, u 2  u 2
z
θ
+
=1
(3)
2A
2αkA
this equation demonstrates that the points on the vibrator surface
exhibit elliptical motions, which drive the rotor via frictional
force. Recently, lightweight USMs have become increasingly
demanded as potential key actuators in robotic applications or
optical instruments [5], [6]. In previous studies, we attempted
to exploit poly phenylene sulfide (PPS), a functional polymer
with low mechanical loss [13]–[15], as the vibrating body to
reduce the USM weight. Compared to the USMs with metalbased vibrators, PPS-based USMs had relatively high rotational
speeds but limited output torques [14].
The bending mode shown in Fig. 1(b) has been employedin our previous polymer-based ring-shaped USMs, as well as
in many conventional metal-based USMs [1]. Various higher
order circumferential modes have been extensively studied in
the previous reports regarding ring- or disk-shaped USMs [9],[10]. However, high-order vibration modes along the verticaldirections have rarely been reported. This study focuses on ahigh-order symmetric vibration mode in the vertical direction.
As depicted in Fig. 1(c), the high-order mode has six vertical nodal lines, which also exist in the conventional third-order1083-4435 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

WU et al.: POLYMER-BASED ULTRASONIC MOTORS UTILIZING HIGH-ORDER VIBRATION MODES

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Fig. 1. Low- and high-order vibration modes in a PPS-based cylindrical vibrator and their admittance characteristics. (a) Structure of a vibrator,and (b) its admittance characteristics measured from 40 Hz to 40 kHz. (b) and (c) Show the vertical vibration velocity distributions in the (0, 3) and(1, 3) modes, respectively. As (d) shows, when a traveling wave in the (0, 3) mode propagates rightwards, the points located on both the upper andback surfaces have counter-clockwise elliptical motions. In contrast, the elliptical motions of points on the upper and back surfaces are anticlockwise
and clockwise with the (1, 3) mode, respectively, as shown in (e). The admittance characteristics of the vibrator measured from 0.04 kHz to 40 kHz
are shown in (f). In the ranges of 11.3–11.5 kHz, and 25.8–26.4 kHz, there are two observable responses, which corresponds to the (0, 3) and(1, 3) modes shown in (g) and (h), respectively. fr and fa represent the resonance and anti-resonance frequencies, respectively, using which, theelectromechanical coupling factor can be calculated.

bending mode, but a horizontal nodal line on the circumferential
outer surface of the vibrator. Many conventional traveling-wave
USMs with ring or hollow-cylindrical shapes exhibit a significant deterioration in performance when their outer surfaces
are fixed because there are no nodes with bending modes [1],[9], [16]. This problem may be solved by using the horizontal
nodal line [17]. In addition, when the vibrator operates in thebending mode, as Fig. 1(d) shows, the points located on boththe upper and back surfaces exhibit elliptical motions in identical directions. In contrast, when the vibrator is operated in thehigh-order mode, as Fig. 1(e) illustrates, the elliptical motionsof the points on its upper and back surfaces have inversely rotational directions [18]. The admittance characteristics of the
PPS-based vibrator shown in Fig. 1(f)–(h) demonstrate that thethird bending and high-order modes are excited at approximately

11.7 and 26.0 kHz, respectively. The high-order mode has anelectromechanical coupling factor of 1.4% (see Section III-B for
its calculation method), relatively high compared to 0.4%, thevalue corresponding to the third bending mode, and it may result in relatively high output power of the PPS-based USM [19].
Here, we use (m, n) to represent a vibration mode, where m and n
are respectively the orders of the vibration modes along the horizontal and vertical directions. Accordingly, the third bending
mode and the high-order mode are referred to as the (0, 3) and (1,3) modes, respectively [18]. Though the vibration performanceof the (1, 3) mode was evaluated in a metal hollow rod [20], its
application to USMs was not systematically examined [21].
In this study, first, we investigate the basic vibration characteristics of the polymer-based vibrators with the (1, 3) modes.
Subsequently, we assess the performance of the motors with

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Fig. 2 Schematic of PPS-based USM. (a) Vibrating body, (b) annular piezoelectric ceramic disk, and (c) rotor. vz , v θ , and v r denote the vibration
velocities along the vertical (z), circumferential (θ), and radial (r) directions, respectively. They form an elliptical trajectory when a traveling wave
propagates on the vibrator.

TABLE IMECHANICAL CONSTANTS
Elastic modulus (GPa) Density (× 103 kg/m3 ) Poisson’s ratio

Material
PPS
Stainless steel

3.45
197

1.35
7.9

0.36
0.29

the (1, 3) modes and make comparisons with other motors.
Finally, we discuss why the (1, 3) and (0, 3) modes are optimal
for polymer-based and thin metal-based motors, respectively,on the basis of their basic vibration characteristics.

Fig. 3. Experimental setup for measuring z-axis vibration velocity distributions along circumferential and top surfaces. LDV: laser Doppler
vibrometer.

II. CONFIGURATION OF PPS-BASED VIBRATOR
Fig. 2(a) illustrates the detailed configuration of the vibrator,
which incorporates a PPS vibrating body and an annular piezoelectric ceramic disk. The vibrating body comprises a cylindricalpart with outer and inner diameters of 30 and 20 mm, respectively, and a back disk with a thickness of 0.5 mm. Slots with
4-mm depths and 0.5-mm widths are introduced at 10° intervals
on the cylindrical part. The thickness of the unslotted part, t, isvaried, and its effects on the motor performance are explored in
the following sections. A central hole is created on the back disk
to fix the vibrator. The mechanical constants of PPS are summarized in Table I. A polar coordinate system (z-, θ-, and r-axes)
is established on the upper surface. The piezoelectric ceramic
disk (C213, Fuji Ceramics, Fujinomiya, Japan) has outer andinner diameters of 20 and 30 mm, respectively, and a thicknessof 0.5 mm. One side of the sliver electrode is evenly divided into
12 parts, of which the polarization directions are identical andthe undivided side is glued onto the back surface of the vibrating
body with epoxy resin. As shown in Fig. 2(b), four channels of
voltages with phase difference of 90° are applied to the electrodes. Thus, along the circumferential direction, the third-order
bending mode is excited. The rotor depicted in Fig. 2(c) consistsof a thin contact rim (30-mm outer and 28-mm inner diameters)
and a flywheel (9-mm thickness and 50-mm diameter). In thisstudy, the vibration velocity distribution on the upper surfaceof the vibrator is not considered because the rotor is in contact
with the outer edge.
III. (1, 3) VIBRATION MODE
A. Vibration Velocity Distribution
First, two voltages with a phase difference of 180°, amplitudesof 50 V, and frequency of 26.35 kHz were applied to generate astanding wave in the (1, 3) mode on the PPS-based vibrator witha 7-mm-thick unslotted part (t7 vibrator). As shown in Fig. 3,
the z-axis vibration velocity was scanned along the vertical direction using an in-plane laser Doppler vibrometer (IPV100,
Polytec, Waldbronn, Germany), and its amplitude A and thephase φ between the vibration velocity and input signal (as areference signal) were recorded with a lock-in voltmeter (5560,
NF Electronic Instruments, Yokohama, Japan). The samplinginterval was set to 1 mm. The z-axis vibration velocity distribution on the outer edge of the upper surface was scanned along
the circumference using an out-of-plane laser Doppler vibrometer (CLV1000, Polytec, Waldbronn, Germany). The samplinginterval was set to 10° (one point per tooth). Fig. 4(a) illustrates
that, as z ranged from 0 to –5 mm, the amplitude of the z-axis
vibration velocity decreased from 52 to 3 mm/s and the phases
were approximately 15°, while the amplitude increased from

WU et al.: POLYMER-BASED ULTRASONIC MOTORS UTILIZING HIGH-ORDER VIBRATION MODES

Fig. 4.

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Distributions of z-axis vibration velocities on the (a) circumferential outer surface and (b) upper surface.

TABLE IIVIBRATION PERFORMANCE FOR VARIOUS HOLDING POSITIONS

Fig. 5. Testbed for measuring admittance characteristics of vibrator
with varying holding position.

3 to 25 mm/s, and the phases were –160° as z was decreased
from –5 to –11 mm. Fig. 4(b) illustrates that three crests andthree troughs appear on the outer edge of the upper surface.
Clearly, the (1, 3) mode is successfully excited on the vibrator. Subsequently, to verify whether the vibrator can be fixedwith low vibration loss by clamping its outer surface, we assessed the vibrator performance when the holding position zH
was changed. Fig. 5 depicts the testbed. The outer surface wastightly held with six bolts that are evenly distributed along thecircumference of a stainless steel holder. From the admittancecharacteristics measured with an impedance analyzer (4294A,
Agilent, Santa Clara, CA, USA), the mechanical quality factor(Q factor) is estimated using the following equation [15], [23] :
Q=

fr
Δf

(4)

where fr and Δf represent the resonance frequency and the
bandwidth corresponding to 0.707 times the peak amplitude,
respectively. Table II lists the results. Clearly, the resonancefrequency variation and the Q-factor reduction at zH = −5 mm,
where the horizontal nodal line is located, are smaller than thevalues at zH = −2 and –10 mm.

Holding position,zH (mm)Resonance frequency,
fr (kHz)

Mechanical quality
factor, Q

Without holder
–2
–5
–10

25.85725.43325.83325.244

182
64
159
60

B. Changes in (1, 3) Vibration Mode With Thickness of
Unslotted Part
By implementing finite element analysis (FEA), we explored
how the (1, 3) mode changed as the thickness of the unslottedpart was varied. Fig. 6(a)–(c) demonstrate the z-axis vibration
velocity distributions on the t3, t7, and t10 vibrators, respectively. The horizontal nodal curve appears on the back disk ofthe t3 vibrator, and on the circumferential outer surface of thet7 vibrator. As depicted in Fig. 6(d), the bilateral sides of theteeth near the vertical nodal lines vibrate out of phase by 180°.
Conversely, in the case of the t10 vibrator, the horizontal nodalline disappears, and all teeth yield out-of-phase vibration velocities. To verify whether these vibration modes were excitedon teeth, the vibration velocity distributions of the t8 and t10
vibrators were measured in a period along the circumference(120°). The voltages applied to the vibrators were 150 V. Here,the sampling interval was set to approximately 1.6° (i.e., five
points per tooth). The vibration velocity vz plotted in Fig. 7 iscalculated from its amplitude A and phase φ as follows:
vz = A · sgn [tan (φ)](5)

where sgn donates the sign function. Fig. 7(a) shows that acosine waveform was generated on the t8 vibrator, and amongthe 12 scanned teeth, the out-of-phase vibration velocities wereobserved on 6 teeth. When the unslotted part thickness increases
to 10 mm, as Fig. 7(b) shows, out-of-phase vibration velocities
are observed on all teeth. This mode is undesirable for USMs.

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Fig. 6. (1, 3) bending modes on PPS-based vibrators with (a) 3-, (b) 7-, and (c) 10-mm-thick unslotted parts. The black dashed curves are thenodal lines located on the circumferential outer surfaces and the bottom surfaces. (d) shows a small scale view of the teeth near the top surface,where 180° out-of-phase vibration velocities are observed. The color legends indicate the z-axis vibration velocities.

Fig. 7. Circumferential z-axis vibration velocity distributions on top surfaces of (a) t8 and (b) t10 vibrators. The dots represent the measured
vibration velocities, and the lines represent the normalized vibration velocities calculated by FEA. The red and blue blocks represent the teeth withand without 180° out-of-phase vibration velocities, respectively.

Several points on the upper surface of the vibrator with relatively
high vibration velocities drive the rotor, but most of the points,
particularly those vibrating out of phase by 180°, generate slips
with the rotor, which lead to friction loss and consequently low
output torque and power [10]. The radius of the nodal circle,
position of the nodal line, and ratio of the number of the teeth
with 180° out-of-phase vibration velocities to the total number ofteeth (nin / ntot ) are plotted in Fig. 8 as functions of the unslottedpart thickness. When the thickness of the unslotted part is lessthan 3.8 mm, a nodal circle, of which the radius increases as thevibrator becomes thicker, is located on the back surface. Whenthe thickness ranges between 3.8 and 8.0 mm, the horizontal
nodal line is located on the circumferential outer surface, andadvances upward along the z-axis. The ratio nin / nout increased
from 50% to 100% as the unslotted part thickness varied fromless than 8 to 10 mm.

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793

Fig. 8. (a) Radius of the nodal circle on bottom surface, rN , and position of the horizontal nodal curve, zN , on circumferential outer surface.
(b) Ratio n in / n to t as a function of unslotted part thickness of the PPS-based vibrator with the (1, 3) mode.

Next, we investigated the basic vibration characteristics ofthe vibrators with varying thickness. The effective electrome2
is estimated from the resonancechanical coupling factor keﬀ
frequency fr and the antiresonance frequency fa [18], [22] asfollows:
 2
fr
2
=1−
.
(6)
keﬀ
fa
The vibration velocity ratios, vθ /vz and vr /vz , show the elliptical motion shapes, which are important indicators for USMs
[1], [2]. Here, vz , vθ , and vr were measured at the antinode,
node, and antinode, respectively. The force factor is defined as
the ratio of the current to the vibration velocity, or equivalently
the ratio of the generated force to the driving voltage [4], and it
significantly affects the output torques of USMs [2], [24]. In thisstudy, we investigated the ratio i/vθ as the force factor becausevθ is parallel to the rotational direction. As Fig. 9(a) shows,
the resonance frequency monotonically decreases from 27.15 to
25.91 kHz, and the electromechanical coupling factor increases
from 1.3% to 1.5% as the unslotted part thickness increases
from 5 to 10 mm. Fig. 9(b) shows that the ratio vθ /vz exhibits
negligible variation, whereas the radial vibration components(vr /vz ), which never contribute to the rotation of USMs [4],[17], increase rapidly. The force factor, as shown in Fig. 9(c),becomes higher with increasing thickness.
IV. MOTOR PERFORMANCE
A. Case Study
The testbed for estimating the motor performance is illustrated in Fig. 10 [4]. The vibrator was clamped to a stainlesssteel shaft fixed to a stainless-steel base plate. A pivot bearing
composed of a conical head and a dent was used to decrease thefriction loss between the shaft and the rotor. The cone angles ofthe bearing head and the dent were 55° and 60°, respectively.
The upper shaft of the bearing was inserted into a Teflon sleeve.

The dent was fixed at the center of the rotor. The preload to therotor was adjusted by varying the weights loaded on the top of
the shaft.
Fig. 11 shows an example to demonstrate how the motor
performance is estimated by the transient method [24], [25].
1) When the voltages were applied to the motor, the transient response of the rotational speed was measured using the two laser Doppler vibrometers [26]. Two beams
illuminated a single point on the outer surface at identical angles, and the rotational velocity was calculated
from the vibrometer outputs. The gray curve shown in
Fig. 11(a) illustrates the transient response of the motorwith the t8 vibrator when voltages of 250 V and a preloadof 4.1 N are applied. After driving voltages were applied at approximately 8 ms, the rotational speed initiallyyielded a rapid increase, but the rotational acceleration
decreased. At approximately 50 ms, the rotational speed
gradually approached saturation, and then began fluctuating because of irregularities in assembly. The cutoff frequencies of the vibrometers were set to 0.2 and 100 kHz.
Thus, the obtained curve transient response contained
the ultrasonic vibration component, which should be thehigh-frequency fluctuation in this curve.
2) Exponential fitting (dashed curve) has been commonly
implemented in previous studies [24], [25], assuming thatrotational speeds of motors linearly decrease as their output torques become higher. However, in general, the load
characteristics of motors are not perfectly linear. Thus,
we employed the polynomial fitting ϕ(t) [see the solid
curve in Fig. 11(a)] to accurately represent the rise curveof the transient response.
3) The instantaneous output torque and power were respectively calculated using the following equation:
T (t) = J ·d [ϕ(t)]
dt

(7)

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Fig. 9. Basic vibration characteristics of polymer-based vibrators with (1, 3) modes. (a) Resonance frequency and electromechanical coupling
factor, (b) vibration velocity ratios, and (c) force factor as functions of unslotted part thickness.

Fig. 10. Testbed for estimating motor performance: (a) Main body;(b) pivot bearing; and (c) rotor and flywheel.

andP (t) = J ·d [ϕ(t)]· ϕ(t)
dt(8)

where J denotes the moment of inertia of the aluminum rotor,
and equals 1.24 × 10−5 kg·m2 . The load characteristics shownin Fig. 11(b) were obtained from the rise part of the transient
response. The maximum torque reached 4.5 mNm when the rotational speed equaled zero, and the maximum rotational speed
was 7.7 rad/s when the rotor stopped accelerating. A maximum
output power of approximately 10.6 mW was achieved whenthe motor yielded a moderate torque.
Fig. 12 demonstrates the motor performance with the (1, 3)
mode as a function of the driving voltage when different preloads
are applied to the motor with the t8 vibrator. Fig. 12(a) showed
that, when the driving voltage was below 100 V, there were deadregions because of friction loss. Then, the maximum torqueincreased proportionally with increasing voltage, and leveled
off at a certain voltage because it was limited by the preload [2],[4]. The maximum torque at 250 V reached 9.2 mNm when an
optimal preload of 9.1 N was applied. As shown in Fig. 12(b),the no-load speed was approximately 12 rad/s at 250 V with apreload of 0.7 N. Fig. 12(c) demonstrated that, as the preload
increased, the output power initially increased owing to theWU et al.: POLYMER-BASED ULTRASONIC MOTORS UTILIZING HIGH-ORDER VIBRATION MODES

795

Fig. 11. (a) Transient response (gray curve) of PPS-based motor with t8 vibrator at voltages of 250 V voltages and a preload of 4.1 N. By
polynomially fitting the rise part of the transient response curve. (b) The load characteristics are obtained.

increasing output torque, and then decreased when the preload
increased beyond a certain value because of the reduction in
the rotation speed. The motor yielded its maximum power of
10.9 mW when an optimal preload 6.4 N was applied to the t8
vibrator.
B. Performance of Motors With Varying Unslotted Part
Thickness
Fig. 13(a)–(c) show the maximum torques, no-load speeds,
and output powers and efficiencies, respectively, when voltagesof 250 V are applied to motors with varying unslotted partthicknesses. Their performance was assessed at three random
different positions. When the unslotted part thickness ranged
from 5 to 7 mm, the maximum torque became higher because of
the increase in the force factor [2], [4]. The no-load speeds havenegligible variation because the tangential vibration velocitiesof the vibrators have almost no changes [10]. The output power
became higher owing to the increasing electromechanical coupling factor. The motor with the t7 vibrator exhibited the highest
output torque, power, and efficiency. When the unslotted partthickness exceeded 7 mm, the output torque, no-load speed, andoutput power exhibited rapid reductions, which can be qualitatively explained from the vibration characteristics. Owing to
the aforementioned tangential vibration components opposing
the rotational direction and the radial vibration components, thefrictional loss tends to be high when the motors have thicker
vibrators. Its negative effect on the output power offsets the
positive effect caused by the increasing electromechanical coupling factors. Thus, the motor performance deteriorates whenthe vibrators are thicker than 7 mm.
Table III compares the performance of the PPS-based motors
with (1, 3) and (0, 3) modes. The motor operating in the (1, 3)
mode has a higher output torque, and the optimal thickness of its
vibrator is lower than that with the (0, 3) mode. A relatively highrotational speed is obtained on the motor with a thin vibrator

when the (0, 3) mode is excited. Owing to the higher electromechanical coupling factor, the motor operating in the (1, 3) mode
has the maximum output power of 14.5 mW, 4.8 times the value
with the (0, 3) mode. The power density, defined as the ratioof the output power to the vibrator weight, is also higher forthe (1, 3) mode. As predicted, higher output torques and powers are achieved in the polymer-based motors by utilizing thehigh-order mode. However, the motors operating in the (1, 3)
modes exhibited relatively low efficiencies, which are assumed
to be dominantly caused by the aforementioned 180° out-ofphase vibration velocities. Note that, owing to the low stiffness,
the teeth easily generate bending vibration (while the unslottedpart does not oscillate), lowering the efficiencies of motors particularly when they have large depths. However, the FEA results
demonstrate that, in the cases of our tested motors, the resonancefrequencies corresponding to the bending vibration on the slots
are in the range 22–24 kHz, deviating from the values for the (1,3) modes. It allows us to infer that the tooth vibration has negligible influence on the motor performance. Additionally, ourpreviously tested polymer-based vibrators with the low-ordermodes were operable in an audible frequency range owing to
low elastic moduli of polymers [15]. By utilizing the high-ordermode, the noise problem has been solved.
C. Comparison Between Polymer- and Metal-Based
Motors Working With (1, 3) ModesFirst, for comparison, the basic vibration characteristics ofstainless-steel-based cylindrical vibrators with the (1, 3) modes
were investigated through FEA. The adequate unslotted partthickness to generate a horizontal nodal line on the circumferential outer surface of the stainless-steel-based vibrator is 15 mm,
which exceeds the practically applicable thickness of cylindrical vibrators [1], [10]. Two stainless-steel-based vibrators with
4- and 15-mm-thick unslotted parts (st4 and st15 vibrators, respectively) were fabricated.

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Fig. 12. Performance of USM with the t8 vibrator. (a) Maximum torque, (b) maximum no-load speed, and (c) maximum power versus driving
voltage under different preloads.

The equivalent electrical circuit shown in Fig. 14 is a typicalmodel for indicating the vibration characteristics [1], [2], [19],[23]. The circuit parameters of the polymer-based t5 and t7 vibrators and stainless-steel-based st4 and st15 vibrators are listed
in Table IV. Clearly, when operating in the (1, 3) modes, thePPS-based vibrators exhibit relatively low equivalent masses
and stiffnesses, resulting in relatively high electromechanicalcoupling factors and consequently high output powers. In contrast, the equivalent stiffness and mass of the stainless-steelbased st4 vibrator are lower when it operates in the (0, 3) mode.
Their optimal modes are different possibly because of the difference in mechanical constants between the vibrating body
(PPS, stainless steel) and excitor (piezoelectric ceramics) [4].
For the st15 vibrator, a lower equivalent stiffness and mass are
obtained with the (1, 3) mode, but they are larger than the valuesof the PPS-based vibrators because of not only higher elastic

modulus and density of stainless steel (see Table I) but also its
high thickness.
Subsequently, the st4 and st15 vibrators are employed to form
motors. To obtain a similar frictional condition, a 0.1-mm-thick
PPS film was glued on the top surfaces. The performance of thestainless-steel-based motors is summarized in Table V. The highelectromechanical coupling factor of the st4 vibrator with the (0,3) mode leads to the high output power of the motor. Comparedto the PPS-based motors with the (0, 3) mode, the motor provides
a relatively high efficiency owing to its low mechanical loss [14],[16], but it cannot rotate with the (1, 3) mode. Though the motorwith the st15 vibrator can rotate with the (1, 3) mode, it yields
a relatively poor performance even when compared to the PPSbased motors. Undesirable tangential vibration components arealso observed on all teeth, and the radial vibration component
is very large for the st15 vibrator.

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Fig. 13. Performance of USMs with varying unslotted part thickness. (a) Maximum torques at 250 V when optimal preloads are applied to vibrators,(b) no-load speeds under minimal preloads and tangential vibration velocities, and (c) maximum output powers and efficiencies.

TABLE IIIPOLYMER-BASED MOTOR PERFORMANCE WITH (1, 3) AND (0, 3) MODES
Performance at 250 V
Driving frequency (kHz)Maximum torque (mNm)
Unslotted part thickness for maximum
torque (mm)Maximum rotational speed (rad/s)
Unslotted part thickness for maximum
rotational speed (mm)Maximum output power (mW)
Unslotted part thickness for maximum
power (mm)Maximum power density (W/kg)
Maximum efficiency(1, 3) mode(0, 3) mode

25–27
9.5
7

7–13
2.9
8

12.3
5–7

16.1
2

14.5
7

3.0
2

5.8
4.0%

2.5
4.8%

Finally, we compared the power densities between ourpolymer-based motors and several traveling-wave USMs reported in the last decade. Table VI indicates that our PPS-based
motor with the (1, 3) mode provides relatively high power density. Additionally, its horizontal nodal line facilitates vibrator

Fig. 14. Simplified equivalent circuit representing a piezoelectric transducer. C d represents the clamped capacitance; and 1/C m , L m , and R mdenote the equivalent stiffness, mass, and damper, respectively.

fixing. With these attractive advantages, our PPS-based motors
are potentially applicable to driving lenses of cameras or optical instruments, where lightweights are required and moderate
output power (or torques) are sufficient, but the fixing problem
remains to be solved, though our current motors cannot provide
better performance than some of the traveling-wave motors operating in more powerful modes [27]. It is also worth mentioning
that, since PPS provides excellent chemical resistance [32], the798

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 23, NO. 2, APRIL 2018

TABLE IV
MEASURED PARAMETERS OF EQUIVALENT CIRCUIT
Vibrator

PPS-based t5 vibrator
PPS-based t7 vibrator
Stainless-steel-based st4 vibrator
Stainless-steel-based st15 vibrator

Vibration
mode

Clamped capacitance,
C d (× 10−12 F)(1, 3)(0, 3)(1, 3)(0, 3)(1, 3)(0, 3)(1, 3)(0, 3)

0.677
0.677
0.676
0.668
0.681
0.681
0.684
0.683

Equivalent stiffness,
1/C m ( × 1012 F −1 )Equivalent mass,
L m (H)

Equivalent damper,
R m (× 103 Ω)

7.4
9.1
5.0
10.1
182.3
0.5
61.2
102.4

271.1
1963.6
188.1
1875.21039.019.3
181.9
1970.5

249.7
591.2
163.6
73.3
970.9
6.6
224.1
668.9

TABLE VPERFORMANCE OF STAINLESS-STEEL-BASED VIBRATORS WITH (1, 3) AND (0, 3) MODES
Indicator

Vibrator performance
Motor performance at 250 V

Driving frequency (kHz)Electromechanical coupling factor
Maximum torque (mNm)
Maximum rotational speed (rad/s)
Maximum output power (mW)Power density (W/kg)
Maximum efficiency

Vibrator with 4-mm-thick unslotted part

Vibrator with 15-mm-thick unslotted part

(1, 3) mode(0, 3) mode

(1, 3) mode(0, 3) mode

66.661
0.1%
Stopped25.956
4.9%
20.8
10.8
31.6
2.8
5.6%

92.293
0.5%
0.8
1.4
0.4
<0.1
0.7%

36.2820.2%
StoppedTABLE VI
POWER DENSITIES OF RECENTLY REPORTED TRAVELING-WAVE USMS
Reference

Material ofvibrating body[27]
Stainless steel
Our polymerPPS
based motor[28]
Stainless steel[29]
Stainless steel[8]
Piezoelectric ceramics[30]
Phosphorus bronze[31]
Stainless steel

Excitation mode inPowerpiezoelectric ceramics density (W/kg)
d 33 (longitudinal)d 31 (traverse)

8.8
5.8

d 31
d 31
d 15 (shear)
d 31
d 31

5.4
4.8
4.3
3.1
2.4

chemical industry would be a promising application area for our
PPS-based motors.
V. CONCLUSION
This study presents the first report of the polymer-based
USMs utilizing the (1, 3) mode. Through an investigation on
the vibration characteristics and motor performance, we havedrawn the following conclusions.
1) The (1, 3) mode introduced six vertical nodal lines and ahorizontal nodal line on the circumferential outer surfaceof the polymer-based vibrator. It offers the advantage offixing the vibrator without excessive vibration loss.
2) The PPS-based vibrator yields a higher electromechanicalcoupling factor with the (1, 3) mode than with the (0, 3)
mode.
3) The optimal thickness of the PPS-based motor is lower
with the (1, 3) mode than with the (0, 3) mode.

4) The maximum torque and power of the PPS-based motor
utilizing the (1, 3) mode reached 9.5 mNm and 14.5 mW,
respectively, higher than the values of both PPS-based
motor using the (0, 3) mode and the stainless-steel-based
motor using the (1, 3) mode.
5) The optimal modes for the PPS-based vibrators with practical thicknesses and stainless-steel-based thin vibrators
are the (1, 3) and (0, 3) modes, respectively, because theirequivalent stiffnesses and masses are relatively low whenworking in the optimal modes.
We have gained an understanding on the polymer-based
USMs with the (1, 3) modes. In the future, the optimal designof the structures of vibrators, rotors, and accessory components
will be investigated. In addition, the vibration modes will be explored in a wider frequency range to find the optimal ones (bothin circumferential and vertical directions) to achieve practicalapplications of our polymer-based USMs.
ACKNOWLEDGMENT
The authors would like to thank the Daicel Corporation forproviding the functional polymers, and the staff members of
the Precision and Manufacturing Center, Technical Department,
Tokyo Institute of Technology, for machining the motor components tested in this study.
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Jiang Wu was born in Liaoning, China, on January 29, 1988. He received the B.E. degree inmechanical engineering from the Dalian University of Technology, Dalian, China, in 2010,
the M.E. degree in mechatronic engineeringfrom the State Key Laboratory of Robotics andSystem, Harbin Institute of Technology, Harbin,
China, in 2012, and the Dr. Eng. degree in electrical and electronic engineering from the Future Interdisciplinary Research of Science andTechnology, Tokyo Institute of Technology, Yokohama, Japan, in 2017.
His research interests include piezoresistive and piezoelectric materials, permanent-magnet-based sensing technology, and polymer-based
ultrasonic transducers and actuators.
Yosuke Mizuno (M’14–SM’17) was born in
Hyogo, Japan, on October 13, 1982. He received
the B.E., M.E., and Dr. Eng. degrees in electronicengineering from the University of Tokyo, Tokyo,
Japan, in 2005, 2007, and 2010, respectively.
From 2007 to 2010, he was involved in Brillouin optical correlation-domain reflectometry for
his Dr.Eng. degree at the University of Tokyo.
From 2007 to 2010, he was a Research Fellow(DC1) of the Japan Society for the Promotion of
Science (JSPS). From 2010 to 2012, as a Research Fellow (PD) of JSPS, he worked on polymer optics with the Tokyo
Institute of Technology, Yokohama, Japan. Since 2012, he has been an
Assistant Professor with the Precision and Intelligence Laboratory (currently the Institute of Innovative Research), Tokyo Institute of Technology,
Yokohama, Japan, where he is active in fiber-optic sensing, polymer optics, and ultrasonics.
Dr. Mizuno was the recipient of the Ando Incentive Prize for the Studyof Electronics in 2011, the Tokyo Tech Challenging Research Awardin 2013, the Konica Minolta Imaging Science Award in 2014, and theJapanese Society of Applied Physics (JSAP) Young Scientist Presentation Award in 2015. He is a member of the IEEE Photonics Society(Senior Member), JSAP, and the Institute of Electronics, Information,
and Communication Engineers of Japan.
Kentaro Nakamura (M’00) was born in Tokyo,
Japan, on July 3, 1963. He received the B.E.,
M.E., and Dr.Eng. degrees from the Tokyo Institute of Technology, Yokohama, Japan, in 1987,
1989, and 1992, respectively.
Since 2010, he has been a Professor with thePrecision and Intelligence Laboratory (currently
the Institute of Innovative Research), Tokyo Institute of Technology. His research interests include applications of ultrasonic waves, measurement of vibration and sound using optical methods, and fiber-optic sensing.
Prof. Nakamura was the recipient of the Awaya Kiyoshi Award for
Encouragement of Research from the Acoustical Society of Japan (ASJ)
in 1996, and the Best Paper Awards from the Institute of Electronics,
Information, and Communication Engineers (IEICE) of Japan in 1998
and from the Symposium on Ultrasonic Electronics in 2007 and 2011.
He also received the Japanese Journal of Applied Physics Editorial Contribution Award from the Japan Society of Applied Physics in 2007. He
is a member of the ASJ, the Japanese Society of Applied Physics, theIEICE, and the Institute of Electrical Engineers of Japan.